Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Transmission Line Resonators

S. R. Zinkazinka@vit.ac.in

School of Electronics EngineeringVellore Institute of Technology

April 29, 2013

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Outline

1 Implementation of Capacitors and Inductors

2 Series and Parallel Resonators

3 Implementation of Resonators using TX Lines

4 Summary

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Outline

1 Implementation of Capacitors and Inductors

2 Series and Parallel Resonators

3 Implementation of Resonators using TX Lines

4 Summary

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Using Short Circuited TX Line

tangent

Inductor

CapacitorCapacitor

Inductor

+

-

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Using Open Circuited TX Line

Inductor Inductor

CapacitorCapacitor

+

-

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Outline

1 Implementation of Capacitors and Inductors

2 Series and Parallel Resonators

3 Implementation of Resonators using TX Lines

4 Summary

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Series and Parallel Resonators

(Series Resonator) (Parallel Resonator)

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Series ResonatorFor the series resonator shown in Fig. X,

Zin = R + jωL +1

jωC.

If it is a loss-less series resonator, R = 0. In either case, resonance occurs when imaginary componentof the impedance becomes zero. So, resonant frequencya

ω0 is given by

jω0L +1

jω0C= 0

⇒ ω0=1√LC

. (1)

One more important definition for resonators is quality factor, which is defined as

Q0 =ω0L

R=

1ω0CR

. (2)

For ideal loss-less resonators quality factor tends to infinity.

aAnother definition of resonant frequency could be the frequency at which stored electric andmagnetic energies are equal.

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Series Resonators - Approximation of Inputimpedance

We have seen that the input impedance of a series resonator is given as

Zin = R + jωL +1

jωC.

This equation can be approximated within the neighborhood of resonant frequency ω0 as shownbelow:

Zin = R + jωL +1

jωC= R + j (ω0 +4ω) L +

1j (ω0 +4ω)C

= R + jω0L + j4ωL +

(1 + 4ω

ω0

)−1

jω0C

≈ R + j4ωL + jω0L +1

jω0C︸ ︷︷ ︸this term is zero

−4ωω0

jω0C= R + j4ωL−

4ωω0

jω0C= R + j

4ω

ω0ω0L−

4ωω0

jω0C

= R[

1 + j4ω

ω0

ω0LR

+ j4ω

ω0

1ω0CR

]= R

[1 + j

4ω

ω0Q0 + j

4ω

ω0Q0

]= R

[1 + j2

4ω

ω0Q0

](3)

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Parallel ResonatorFor the parallel resonator shown in Fig. X,

Yin =1R

+1

jωL+ jωC.

If it is a loss-less parallel resonator, R → ∞. In either case, resonance occurs when imaginary com-ponent of the admittance becomes zero. So, resonant frequencya

ω0 is given by

1jω0L

+ jω0C = 0

⇒ ω0=1√LC

. (4)

One more important definition for resonators is quality factor, which is defined as

Q0 =R

ω0L= ω0CR. (5)

For ideal loss-less resonators quality factor tends to infinity.

aAnother definition of resonant frequency could be the frequency at which stored electric andmagnetic energies are equal.

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Parallel Resonators - Approximation of Inputimpedance

We have seen that the input admittance of a parallel resonator is given as

Yin =1R

+1

jωL+ jωC.

So, the equation for input impedance can be approximated within the neighborhood of resonantfrequency ω0 as shown below:

Zin =

[1R

+1

jωL+ jωC

]−1

=

[1R

+1

j (ω0 +4ω) L+ j (ω0 +4ω)C

]−1

=

1R

+

(1 + 4ω

ω0

)−1

jω0L+ jω0C + j4ωC

−1

≈

1R

+

(1− 4ω

ω0

)jω0L

+ jω0C + j4ωC

−1

=

1R−

4ωω0

jω0L+

1jω0L

+ jω0C + j4ωC

−1

=

1R

1−R4ω

ω0jω0L

+ j4ω

ω0ω0CR

−1

= R[

1 + jR

ω0L4ω

ω0+ jω0CR

4ω

ω0

]−1

= R[

1 + jQ04ω

ω0+ jQ0

4ω

ω0

]−1

=R

1 + j2Q04ωω0

(6)

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Outline

1 Implementation of Capacitors and Inductors

2 Series and Parallel Resonators

3 Implementation of Resonators using TX Lines

4 Summary

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Simple Explanation (Assuming Loss-less TX Lines)

CASE 1

CASE 3

CASE 2

CASE 4

• For CASE 1 and 4, input impedance Zin = 0 ... So, they both are series resonators

• For CASE 2 and 3, input impedance Zin → ∞ ... So, they both are parallel resonators

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

λ02 Short Circuited TX Line (CASE 4)

This time, it is assumed that the TX line is made up of lossy elements so that α 6= 0. In such case,input impedance is given as

Zin = Z0

[ZL + Z0 tanh γlZ0 + ZL tanh γl

]= Z0 tanh γl, ∵ ZL = 0

= Z0 tanh (αl + jβl)

= Z0tanh αl + tanh (jβl)

1 + tanh αl tanh (jβl)

≈ Z0αl + j tan βl

1 + jαl tan βl, ∵ tanh αl ≈ αl (7)

Within the neighborhood of center frequency ω0, βl can be written as

βl =2π

λl =

2π

λ

λ0

2= π

ω

ω0= π

(ω0 + ∆ω)

ω0= π

(1 +

∆ω

ω0

).

So,

tan βl = tan(

π∆ω

ω0

)≈ π

∆ω

ω0. (8)

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

λ02 Short Circuited TX Line (CASE 4) ... Contd

Substituting (8) in (7) gives

Zin ≈ Z0

αl + jπ ∆ωω0

1 + jαlπ ∆ωω0

= Z0

(αl + jπ

∆ω

ω0

), ∵ αl

∆ω

ω0� 1 (9)

Comparing the above equation with the input impedance of series resonator Zin ≈ R[1 + j24ω

ω0Q0

]gives

{R = Z0αlQ0 = π

2αl =β02α , since β0l = π

. (10)

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

λ04 Short Circuited TX Line (CASE 3)

Input impedance of a lossy TX line is given as

Zin = Z0

[ZL + Z0 tanh γlZ0 + ZL tanh γl

]= Z0 tanh γl, ∵ ZL = 0

≈ Z0αl + j tan βl

1 + jαl tan βl(11)

= Z0−j cot βlαl + 1−j cot βl + αl

. (after multiplying with −j cot βl) (12)

Within the neighborhood of center frequency ω0, βl can be written as

βl =2π

λl =

2π

λ

λ0

4=

π

2ω

ω0=

π

2(ω0 + ∆ω)

ω0=

π

2

(1 +

∆ω

ω0

).

So,

cot βl = − tan(

π

2∆ω

ω0

)≈ −π

2∆ω

ω0. (13)

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

λ04 Short Circuited TX Line (CASE 3) ... Contd

Substituting (13) in (11) gives

Zin ≈ Z0−j cot βlαl + 1−j cot βl + αl

= Z0

j(

π2

∆ωω0

)αl + 1

j(

π2

∆ωω0

)+ αl

=Z0[

αl + j(

π2

∆ωω0

)] , ∵(

π

2∆ω

ω0

)αl� 1. (14)

Comparing the above equation with the input impedance of parallel resonator Zin ≈ R[1+j24ω

ω0Q0]

gives

{R =

Z0αl

Q0 = π4αl =

β02α , since β0l = π

2

. (15)

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Outline

1 Implementation of Capacitors and Inductors

2 Series and Parallel Resonators

3 Implementation of Resonators using TX Lines

4 Summary

Vector Calculus EE208, School of Electronics Engineering, VIT

Implementation of Capacitors and Inductors Series and Parallel Resonators Implementation of Resonators using TX Lines Summary

Summary

Series Resonators:

• ω0 = 1√LC

• Q0 =ω0L

R = 1ω0CR

• Zin ≈ R[1 + j24ω

ω0Q0

]Parallel Resonators:

• ω0 = 1√LC

• Q0 =ω0L

R = 1ω0CR

• Zin ≈ R[1 + j24ω

ω0Q0

]

λ02 Short Circuited TX Line:

• R = Z0αl

• Q0 = π2αl =

β02α

• Zin ≈ Z0

(αl + jπ ∆ω

ω0

)λ04 Short Circuited TX Line:

• R =Z0αl

• Q0 = π4αl =

β02α

• Zin ≈Z0[

αl+j(

π2

∆ωω0

)]

Vector Calculus EE208, School of Electronics Engineering, VIT